setcover - PCPs and Inapproxiability CIS 6930 October 5,...

PCPs and Inapproxiability CIS 6930 October 5, 2009 Lecture Hardness of Set Cover Lecturer: Dr. My T. Thai Scribe: Ying Xuan 1 Preliminaries 1.1 Two-Prover-One-Round Proof System A new PCP model 2P1R – Think of the proof system as a game between two provers and one veriﬁer. • Prover Model: – each tries to cheat – convince the veriﬁer that a ”No” instance is ”Yes”; – each cannot communicate with the other on their answers; • Veriﬁer Model: – tries not to be cheated – ensures the probability to be cheated is upper bounded by some small constant; – can cross-check the two provers’ answers; – is only allowed to query one position in each of the two proofs. Deﬁnition 1 Given three parameters: completeness c , soundness s and the number of random bits provided to the veriﬁer r ( n ) , assume the two proofs are written in two alphabets Σ 1 and Σ 2 respectively, then a language L is in 2P1R c,s ( r ( n )) if there is a polynomial time bounded veriﬁer V that receives O ( r ( n )) truly random bits and satisﬁes: • for every input x ∈ L , there is a pair of proofs y 1 ∈ Σ * 1 and y 2 ∈ Σ * 2 that makes V accept with probability ≥ c ; • for every input x / ∈ L and every pair of proofs y 1 ∈ Σ * 1 and y 2 ∈ Σ * 2 makes V accept with probability < s ; 2P1R is actually a mechanism to solve Label-Cover: L ( G = ( U,V ; E ) , Σ , Π) ⇔ u ∈ U and v ∈ V , we have • v : proof P 1 ; u : proof P 2 ; • uv ∈ E : u , v have answers for the same bit; • a labeling L is said to satisfy an edge uv iff Π uv ( L ( u )) = L ( v ) : the two answers are consistent, otherwise they are cheating. Hardness of Set Cover-1

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